| OEIS | step set | alg | equiv | |
|---|---|---|---|---|
| 1 | A005568 | N | $\begin{cases} \frac{32 }{\pi } \frac{4 ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 2 | A001246 | N | $\begin{cases} \frac{8 }{\pi } \frac{4 ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 3 | A151362 | N | $\begin{cases} \frac{3 \sqrt{6 }}{\pi } \frac{6 ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 4 | A172361 | N | $\frac{128 }{27 \pi } \frac{8 ^n}{n^{3}}$ | |
| 5 | A151332 | N | $\begin{cases} \frac{16 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p) \\ 0 & (n=4p+1) \\ 0 & (n=4p+2) \\ 0 & (n=4p+3) \end{cases}$ | |
| 6 | A151357 | N | $\frac{2 A ^{3/2} }{\pi } \frac{(2 A ) ^n}{n^{3}}$ | |
| 7 | A151341 | N | $\begin{cases} \frac{12 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 8 | A151368 | N | $\frac{2 B ^{3/2} }{\pi } \frac{(2 B ) ^n}{n^{3}}$ | |
| 9 | A151345 | N | $\begin{cases} \frac{24 \sqrt{30 }}{25 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 10 | A151370 | N | $\frac{2 \mu ^{3} C ^{3/2} }{\pi } \frac{(2 C ) ^n}{n^{3}}$ | |
| 11 | A151341 | N | $\begin{cases} \frac{12 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 12 | A151368 | N | $\frac{2 B ^{3/2} }{\pi } \frac{(2 B ) ^n}{n^{3}}$ | |
| 13 | A151345 | N | $\begin{cases} \frac{24 \sqrt{30 }}{25 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 14 | A151370 | N | $\frac{2 \mu ^{3} C ^{3/2} }{\pi } \frac{(2 C ) ^n}{n^{3}}$ | |
| 15 | A151332 | N | $\begin{cases} \frac{16 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p) \\ 0 & (n=4p+1) \\ 0 & (n=4p+2) \\ 0 & (n=4p+3) \end{cases}$ | |
| 16 | A151357 | N | $\frac{2 A ^{3/2} }{\pi } \frac{(2 A ) ^n}{n^{3}}$ | |
| 17 | A151334 | N | $\begin{cases} \frac{81 \sqrt{3 }}{\pi } \frac{3 ^n}{n^{4}} & (n=3p) \\ 0 & (n=3p+1) \\ 0 & (n=3p+2) \end{cases}$ | |
| 18 | A151366 | N | $\frac{27 \sqrt{3 }}{\pi } \frac{6 ^n}{n^{4}}$ | |
| 19 | A138349 | N | $\begin{cases} \frac{768 }{\pi } \frac{4 ^n}{n^{5}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ |
Nature of the generating series for (vx,vy)=(0,0) and coefficient asymptotics
$A=1+\sqrt2$, \ $B=1+\sqrt3$, \ $C=1+\sqrt6$, \ $\lambda=7+3\sqrt6$, \ $\mu=\sqrt{\frac{4\sqrt6-1}{19}}$
| OEIS | step set | alg | equiv | |
|---|---|---|---|---|
| 1 | A005558 | N | $\frac{8 }{\pi } \frac{4 ^n}{n^{2}}$ | |
| 2 | A151392 | N | $\begin{cases} \frac{4 }{\pi } \frac{4 ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 3 | A151478 | N | $\frac{3 \sqrt{6 }}{2 \pi } \frac{6 ^n}{n^{2}}$ | |
| 4 | A151496 | N | $\frac{32 }{9 \pi } \frac{8 ^n}{n^{2}}$ | |
| 5 | A151380 | N | $\frac{3 }{4 } \sqrt{\frac{3 }{\pi }} \frac{3 ^n}{n^{3/2}}$ | |
| 6 | A151450 | N | $\frac{5 }{16 } \sqrt{\frac{10 }{\pi }} \frac{5 ^n}{n^{3/2}}$ | |
| 7 | A148790 | N | $\frac{8 }{3 \sqrt{\pi }} \frac{4 ^n}{n^{3/2}}$ | |
| 8 | A151485 | N | $\sqrt{\frac{3 }{\pi }} \frac{6 ^n}{n^{3/2}}$ | |
| 9 | A151440 | N | $\frac{5 }{24 } \sqrt{\frac{10 }{\pi }} \frac{5 ^n}{n^{3/2}}$ | |
| 10 | A151493 | N | $\frac{7 }{54 } \sqrt{\frac{21 }{\pi }} \frac{7 ^n}{n^{3/2}}$ | |
| 11 | A151394 | N | $\begin{cases} \frac{36 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{3}} & (n=2p) \\ \frac{54 }{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{3}} & (n=2p+1) \end{cases}$ | |
| 12 | A151472 | N | $\frac{3 B ^{7/2} }{2 \pi } \frac{(2 B ) ^n}{n^{3}}$ | |
| 13 | A151437 | N | $\begin{cases} \frac{72 \sqrt{30 }}{5 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{3}} & (n=2p) \\ \frac{864 \sqrt{5 }}{25 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{3}} & (n=2p+1) \end{cases}$ | |
| 14 | A151492 | N | $\frac{6 \lambda \mu ^{3} C ^{5/2} }{5 \pi } \frac{(2 C ) ^n}{n^{3}}$ | |
| 15 | A151375 | N | $\begin{cases} \frac{448 \sqrt{2 }}{9 \pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p) \\ \frac{640 }{9 \pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p+1) \\ \frac{416 \sqrt{2 }}{9 \pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p+2) \\ \frac{512 }{9 \pi } \frac{(2 \sqrt{2 }) ^n}{n^{3}} & (n=4p+3) \end{cases}$ | |
| 16 | A151430 | N | $\frac{4 A ^{7/2} }{\pi } \frac{(2 A ) ^n}{n^{3}}$ | |
| 17 | A151378 | N | $\frac{27 }{8 } \sqrt{\frac{3 }{\pi }} \frac{3 ^n}{n^{5/2}}$ | |
| 18 | A151483 | Y | $\frac{27 }{8 } \sqrt{\frac{3 }{\pi }} \frac{6 ^n}{n^{5/2}}$ | |
| 19 | A005568 | N | $\begin{cases} \frac{32 }{\pi } \frac{4 ^n}{n^{3}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ |
Nature of the generating series for (vx,vy)=(0,1) and coefficient asymptotics
$A=1+\sqrt2$, \ $B=1+\sqrt3$, \ $C=1+\sqrt6$, \ $\lambda=7+3\sqrt6$, \ $\mu=\sqrt{\frac{4\sqrt6-1}{19}}$
| OEIS | step set | alg | equiv | |
|---|---|---|---|---|
| 1 | A005558 | N | $\frac{8 }{\pi } \frac{4 ^n}{n^{2}}$ | |
| 2 | A151392 | N | $\begin{cases} \frac{4 }{\pi } \frac{4 ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 3 | A151471 | N | $\begin{cases} \frac{2 \sqrt{6 }}{\pi } \frac{6 ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 4 | A151496 | N | $\frac{32 }{9 \pi } \frac{8 ^n}{n^{2}}$ | |
| 5 | A151379 | N | $\begin{cases} \frac{4 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 6 | A148934 | N | $\frac{\sqrt{2 }A ^{3/2} }{\pi } \frac{(2 A ) ^n}{n^{2}}$ | |
| 7 | A151410 | N | $\begin{cases} \frac{4 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 8 | A151464 | N | $\frac{2 B ^{3/2} \sqrt{3 }}{3 \pi } \frac{(2 B ) ^n}{n^{2}}$ | |
| 9 | A151423 | N | $\begin{cases} \frac{4 \sqrt{30 }}{5 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 10 | A151490 | N | $\frac{\sqrt{6 }\mu C ^{3/2} }{3 \pi } \frac{(2 C ) ^n}{n^{2}}$ | |
| 11 | A151410 | N | $\begin{cases} \frac{4 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 12 | A151464 | N | $\frac{2 B ^{3/2} \sqrt{3 }}{3 \pi } \frac{(2 B ) ^n}{n^{2}}$ | |
| 13 | A151423 | N | $\begin{cases} \frac{4 \sqrt{30 }}{5 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 14 | A151490 | N | $\frac{\sqrt{6 }\mu C ^{3/2} }{3 \pi } \frac{(2 C ) ^n}{n^{2}}$ | |
| 15 | A151379 | N | $\begin{cases} \frac{4 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=2p) \\ 0 & (n=2p+1) \end{cases}$ | |
| 16 | A148934 | N | $\frac{\sqrt{2 }A ^{3/2} }{\pi } \frac{(2 A ) ^n}{n^{2}}$ | |
| 17 | A151497 | N | $\frac{27 }{8 } \sqrt{\frac{3 }{\pi }} \frac{3 ^n}{n^{5/2}}$ | |
| 18 | A151483 | Y | $\frac{27 }{8 } \sqrt{\frac{3 }{\pi }} \frac{6 ^n}{n^{5/2}}$ | |
| 19 | A005817 | N | $\frac{32 }{\pi } \frac{4 ^n}{n^{3}}$ |
Nature of the generating series for (vx,vy)=(1,0) and coefficient asymptotics
$A=1+\sqrt2$, \ $B=1+\sqrt3$, \ $C=1+\sqrt6$, \ $\lambda=7+3\sqrt6$, \ $\mu=\sqrt{\frac{4\sqrt6-1}{19}}$
| OEIS | step set | alg | equiv | |
|---|---|---|---|---|
| 1 | A005566 | N | $\frac{4 }{\pi } \frac{4 ^n}{n}$ | |
| 2 | A018224 | N | $\frac{2 }{\pi } \frac{4 ^n}{n}$ | |
| 3 | A151312 | N | $\frac{\sqrt{6 }}{\pi } \frac{6 ^n}{n}$ | |
| 4 | A151331 | N | $\frac{8 }{3 \pi } \frac{8 ^n}{n}$ | |
| 5 | A151266 | N | $\frac1{2 } \sqrt{\frac{3 }{\pi }} \frac{3 ^n}{n^{1/2}}$ | |
| 6 | A151307 | N | $\frac1{4 } \sqrt{\frac{10 }{\pi }} \frac{5 ^n}{n^{1/2}}$ | |
| 7 | A151291 | N | $\frac{4 }{3 \sqrt{\pi }} \frac{4 ^n}{n^{1/2}}$ | |
| 8 | A151326 | N | $\frac{2 }{3 } \sqrt{\frac{3 }{\pi }} \frac{6 ^n}{n^{1/2}}$ | |
| 9 | A151302 | N | $\frac1{6 } \sqrt{\frac{10 }{\pi }} \frac{5 ^n}{n^{1/2}}$ | |
| 10 | A151329 | N | $\frac1{9 } \sqrt{\frac{21 }{\pi }} \frac{7 ^n}{n^{1/2}}$ | |
| 11 | A151261 | N | $\begin{cases} \frac{12 \sqrt{3 }}{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{2}} & (n=2p) \\ \frac{18 }{\pi } \frac{(2 \sqrt{3 }) ^n}{n^{2}} & (n=2p+1) \end{cases}$ | |
| 12 | A151297 | N | $\frac{\sqrt{3 }B ^{7/2} }{2 \pi } \frac{(2 B ) ^n}{n^{2}}$ | |
| 13 | A151275 | N | $\begin{cases} \frac{12 \sqrt{30 }}{\pi } \frac{(2 \sqrt{6 }) ^n}{n^{2}} & (n=2p) \\ \frac{144 \sqrt{5 }}{5 \pi } \frac{(2 \sqrt{6 }) ^n}{n^{2}} & (n=2p+1) \end{cases}$ | |
| 14 | A151314 | N | $\frac{\sqrt{6 }\lambda \mu C ^{5/2} }{5 \pi } \frac{(2 C ) ^n}{n^{2}}$ | |
| 15 | A151255 | N | $\begin{cases} \frac{24 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=4p) \\ \frac{32 }{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=4p+1) \\ \frac{24 \sqrt{2 }}{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=4p+2) \\ \frac{32 }{\pi } \frac{(2 \sqrt{2 }) ^n}{n^{2}} & (n=4p+3) \end{cases}$ | |
| 16 | A151287 | N | $\frac{2 \sqrt{2 }A ^{7/2} }{\pi } \frac{(2 A ) ^n}{n^{2}}$ | |
| 17 | A001006 | Y | $\frac{3 }{2 } \sqrt{\frac{3 }{\pi }} \frac{3 ^n}{n^{3/2}}$ | |
| 18 | A129400 | Y | $\frac{3 }{2 } \sqrt{\frac{3 }{\pi }} \frac{6 ^n}{n^{3/2}}$ | |
| 19 | A005558 | N | $\frac{8 }{\pi } \frac{4 ^n}{n^{2}}$ |
Nature of the generating series for (vx,vy)=(1,1) and coefficient asymptotics
$A=1+\sqrt2$, \ $B=1+\sqrt3$, \ $C=1+\sqrt6$, \ $\lambda=7+3\sqrt6$, \ $\mu=\sqrt{\frac{4\sqrt6-1}{19}}$